**MATHEMATICS of RELATIVITY**

**(OBSERVER’S MATHEMATICS)**

**Boris Khots, Dmitriy Khots**

**Abstract**

This e:book considers authors' results, which were done for so-called Observer's Mathematics.
This mathematics was introduced by authors including research of its various properties and its applications to
classical mathematics itself and to contemporary physics.

When we consider and analyze physical events with the purpose of creating corresponding models we often assume
that the mathematical apparatus used in modeling is infallible. In particular, this relates to the use of infinity in various
aspects and the use of Newton's definition of a limit in analysis. First of all we define Observer's Mathematics which we
have created to avoid infinity idea and consider as an alternative to contemporary mathematics.

Observer dependent ascending chain of embedded sets of decimal fractions and their Cartesian products are considered.
For every set, arithmetic operations are defined (these operations locally coincide with standard operations).
The basic problems of Algebra, Geometry, Topology, and Logic are solved for this chain.

Certain results that have been predicted by contemporary Theoretical Physics are not always supported by experiments.
We prove that, in fact, the mathematical apparatus employed within today's physics is a possible reason.
We consider the Hamiltonian Mechanics, Newton equation, Schrodinger equation, two slit interference,
wave-particle duality for single photons, uncertainty principle, Dirac equations for free electron, photoelectric effect in a setting of arithmetic,
algebra, and topology provided by Observer's Mathematics.

We consider Observer's Mathematics applications to classical mathematics, Special Relativity Theory, General Relativity Theory and Gravitation.
The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's
theory of general relativity. The main tools used in this geometrical theory of gravitation are tensor fields.
The principle of general covariance states that the laws of physics should take the same mathematical form in all reference frames and
was one of the central principles in the development of general relativity. When we go to Observer's Mathematics point of view,
we note immediately that ``tensor idea'' becomes incorrect. I.e. the idea of equality of all coordinate systems (local basises) becomes incorrect.
We proved that tensors of any type in classical Linear Algebra are not tensors in Observer's Mathematics.
They are only tensors with some probability less than 1.

We proved that Mercury perihelion movement is connected with not precession,
but with spatial (not a planar) orbit and changing distance between Sun and this point.

The Math of Relativity was presented by authors on:

July 5 – 6, 2017

Presentation at informal Conference
“Observer’s Mathematics and it’s applications to contemporary Physics and classical Mathematics”, Omaha Nebraska, USA

Discussions on Conference have been around three new books:

Observer's Mathematics applications to Quantum Mechanics are considered in book
''Boris Khots and Dmitriy Khots, Quantum Mechanics from Observer's Mathematics point of view, 132 pp, DOI: 10.12732/acadpublmon2015007, Academic Publications, 2015'';

Observer's Mathematics applications to Relativity theory are considered in book
''Boris Khots and Dmitriy Khots, Special and General Relativity theory and Gravitation from Observer's Mathematics point of view, 120 pp, ISBN 978-5-906818-47-8, KURS Publishing House, 2016''.

Observer's Mathematics and it's applications to Quantum Mechanics, Relativity theory and classical Mathematics are considered in book
''Boris Khots and Dmitriy Khots, Observer's Mathematics and it's applications to Quantum Mechanics, Relativity theory and Classical Mathematics, 160 pp, ISBN 978-5-906923-21-9, ISBN 978-5-16-101348-9 (INFRA-M, online), KURS Publishing House, 2017''

(in Russian language: Б. Хоц и Д. Хоц, Математика наблюдателей и ее приложения к квантовой механике, теории относительности и классической математике).

August 10 – 13, 2015

Presentation at
The SPIE Optics + Photonics 2015 Conference “9570 - The Nature of Light: What are Photons? VI”
, San Diego, USA

March 9 – 11, 2015

Presentation (plenary talk) at
Workshop on "Quantum Probability and the Mathematical Modeling
of Decision Making", Fields Institute, Toronto, Canada

August 12 – 22, 2014

Presentation at
the International Congress of Mathematicians 2014,
"ICM2014" Seoul, South Korea

NOTE: Certain results that have been predicted by Quantum Mechanics (QM) theory are not always
supported by experiments. This defines a deep crisis in contemporary physics and, in particular,
quantum mechanics. We believe that, in fact, the mathematical apparatus employed within today’s physics
is a possible reason. In particular, we consider the concept of infinity that exists in today
mathematics as the root cause of this problem.
We have created Observer’s Mathematics that offers an alternative to contemporary mathematics.
This presentation is an attempt to explain how Observer’s Mathematics may explain some of the contradictions
in QM theory results.

July 14 – 18, 2014

Presentations (keynote talk and section talk) at the
ICNPAA 2014 conference
(world congress), Narvik, Norway

June 9 – 12, 2014

Presentation at
Vaxjo University International Conference
"Quantum Theory from Problems to Advances - QTPA",
Vaxjo, Sweden

September 9 – 12, 2013

Presentation at 8th International Conference “Topology, Geometry and Geometry teaching”, Cherkassy State Technology University, Cherkassy, Ukraine. Authors would like to thank Professor Yury Aminov for the presentation of their talk.

August 25 – 29, 2013

Presentation at the
SPIE Optics + Photonics 2013 Conference “8832 - The Nature of Light: What are Photons? V”, San Diego, USA

June 10 – 13, 2013

Presentation at
Vaxjo University International Conference "Quantum Theory: Advances and Problems - QTAP", Vaxjo, Sweden

June 4 – 9, 2013

Presentation at the
International Conference "Nonlinear Mathematical Physics: Twenty years of JNMP", Nordfjordeid, Norway. Authors would like to thank Professor Norbert Euler for the presentation of their talk

May 25 – 27, 2013

Presentation at
10th HSTAM (The Hellenic Society for Theoretical and Applied Mechanics) 2013 International Congress on Mechanics, Chania, Crete, Greece.

June 11-14, 2012

Presentation at the Vaxjo University International Conference
“Quantum Theory: Reconsideration of Foundations - 6”, Vaxjo, Sweden

June 12-15, 2012

Presentation at the 5th Chaotic Modeling and Simulation International Conference (CHAOS2012),
Athens, Greece. Authors would like to thank Professor Christos H. Skiadas for the presentation of their talk.

July 23, 2012

Presentation at Kazan State University, Department of Gravity and Theory of Relativity (Prof. A. Aminova), Kazan, Russia

July 10-14, 2012

Presentation at the ICNPAA 2012 conference(world congress), Vienna, Austria

November 10-11, 2011

Presentation at “Contemporary information technologies in logistic-transportation control systems” International Conference, International Informatization Akademy, Kazan, Tatarstan, Russia. Authors would like to thank Academic Albert Bekmullin for the presentation of their talk.

August 22-25, 2011

Presentation at “The nature of light: What are Photons? IV” SPIE International Conference, San Diego, California, USA.

June 13-16, 2011

Presentation at “Foundations of Probability and Physics (FPP-6)” International Conference, Vaxjo University, Sweden.

May 29 - June 3, 2011

Presentation at “Chaotic Modeling and Simulation (CHAOS 2011)” International Conference, Agios Nikolaos, Crete Greece.

November 1-6, 2010

Presentation at "The Petrov 2010 Anniversary Symposium on General Relativity and Gravitation" at the Kazan State University, Russia

August 19-27, 2010

Presentation at the International Congress of Mathematicians 2010, "ICM2010" Hyderabad,
India

NOTE: Authors showed that negative solution of the classical 10th
Hilbert problem depends on validity of “Axiom of Choice” in standard
(“naive”) Mathematics. In Mathematics of Relativity this axiom becomes
invalid (authors proved that), and 10th Hilbert problem has positive
solution

August 3-7, 2010

Presentation at "17th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (17th ICFIDCAA)" Ho Chi Min City University of Pedagogy, Vietnam

Authors would like to thank Professor Le Hung Son for the presentation of their talk.

June 1-5, 2010

Presentation at "3rd Chaotic Modeling and Simulation” International Conference," Chania Crete, Greece

September 14-20, 2009

presentation at the Novosibirsk University The International Conference "Contemporary Analysis and Geometry"

Authors would like to thank Professor Sergey Vodopyanov for the presentation of their talk “Geometrical and Analytical aspects of Observer’s Mathematics”.

June 22-27, 2009

presentation at the International Conference “Geometry “in large”, topology and applications”, devoted to the 90th anniversary Alexey Vasilievich Pogorelov,

Kharkov, Ukraine

June 1-5, 2009

presentation at the “2nd Chaotic Modeling and Simulation” International Conference,

Chania Crete, Greece

June 14-18, 2009

presentation at the Vaxjo University International Conference “Quantum Theory: Reconsideration of Foundations - 5”,

Vaxjo, Sweden

August 24-27, 2008

presentation at Vaxjo University Conference
“Foundations of
Probability and Physics- 5”

Vaxjo, Sweden

August 21-26, 2008

presentation at Debrecen University 6th Bolyai – Gauss - Lobachevsky Conference
(BGL6) “Non-Euclidean Geometry and its
Applications”, Debrecen, Hungary

August 12-18, 2008

presentation at Fifth
International Conference of Applied Mathematics and Computing, Plovdiv,
Bulgaria

July 28 – August 1, 2008

presentation at 16th
International Conference on Finite or Infinite Dimensional Complex Analysis and
Applications (16th ICFIDCAA), Dongguk University (Gyeongju), Korea

Authors would like to thank Professor Junesang Choi for the presentation of
their talk.

June 11-16, 2007

presentation at Vaxjo
University Conference “Quantum Theory: Reconsideration of Foundations - 4”,
Vaxjo, Sweden.

May 18-20, 2007

presentation at Midwest Geometry
Conference – 2007, The University of Iowa, Iowa City, Iowa, USA

October 14, 2006

presentation at __
Kazan State University__,
Department of Gravity and Theory or Relativity (Prof. A. Aminova), Kazan, Russia

August 22-30, 2006

presentation at International Congress of Mathematicians 2006,
ICM2006, Madrid,
Spain

NOTE: Authors showed that negative solution of the classical Fermat problem
depends on validity of “Axiom of Choice” in standard (“naive”) Mathematics. In
Mathematics of Relativity this axiom becomes invalid (authors proved that), and
Fermat problem has positive solution – see below.

October 6-8, 2005

presentation at Wolfram Technology Conference 2005,
http://www.wolfram.com/techconf2005, Champaign, Illinois, USA

September 30, 2005

presentation at The University of Iowa,
www.math.uiowa.edu, Department of Mathematics, Topology Seminar (Prof. Jon
Simon), Iowa City, Iowa, USA

August 22-27, 2005

presentation at International Symposium “Analytic Function Theory, Fractional
Calculus and Their Applications”,
http://www.pims.math.ca/science/2005/05hms/program.html, University of
Victoria, Department of Mathematics and Statistics, Victoria, Canada

June 22 – July3, 2005

presentation at the XVIIth Summer School-Seminar VOLGA-2005,
www.ksu.ras.ru , Kazan, Tatarstan, Russia

April 21, 2005

presentation at The University of Iowa,
www.math.uiowa.edu, Department of Mathematics, Mathematical Biology Seminar
(Prof. Herbert Hethcold), Iowa City, Iowa, USA

31st January, 2005

presentation at Higher School of Economics, Department
of Mathematics (Prof. S. Strunkov), Moscow, Russia

2nd December, 2004

presentation at
Sibirian State Geodesics Academy, Department of Higher Mathematics (Prof. I. Vovk),
Novosibirsk, Russia

1st December, 2004

presentation at Novosibirsk State University,
Department of Mathematics (Prof. A. Gutman), Novosibirsk, Russia

30th November, 2004 and 3rd December, 2004

presentations at Novosibirsk State Technical
University, Department of Mathematics (Prof. V. Seleznev), Novosibirsk, Russia

29th November, 2004

presentation at Novosibirsk State University,
Department of Mathematics (Prof. Sergey Krendelev), Novosibirsk, Russia

1st October, 2004

presentation at
Kazan
State University, Department of Gravity and Theory or Relativity (Prof. A. Aminova),
Kazan, Russia

29-30 September, 2004

International Conference "Informatics problems in third millennium", Kazan,
Russia

22 June - 3 July, 2004

THE XVIth
SUMMER SCHOOL-SEMINAR VOLGA-2004, Russia, Tatarstan, Kazan

5 - 10 July, 2004

Conference on Non Standard
Mathematics, Portugal, University of Aveiro

Mathematics of Relativity Publications

'Boris Khots and Dmitriy Khots, Observer's Mathematics and it's applications to Quantum Mechanics, Relativity theory and Classical Mathematics,
160 pp, ISBN 978-5-906923-21-9, ISBN 978-5-16-101348-9 (INFRA-M, online), KURS Publishing House, 2017'

(in Russian language: Б. Хоц и Д. Хоц, Математика наблюдателей и ее приложения к квантовой механике,
теории относительности и классической математике).

'Boris Khots and Dmitriy Khots, Special and General Relativity theory and Gravitation from Observer's Mathematics point of view, 120 pp,
ISBN 978-5-906818-47-8, KURS Publishing House, 2016'.

Boris Khots, Dmitriy Khots, Quantum Mechanics from Observer’s Mathematics point of view, 132 pp,

DOI: 10.12732/acadpublmon2015007, Academic Publications , 2015

Boris Khots, Dmitriy Khots, Special Relativity from Observer’s Mathematics point of view, Proceedings of the SPIE Optics + Photonics 2015 Conference “9570 - The Nature of Light: What are Photons? VI”, vol. 9570, pp 95701E-1 – 9570E-12,

DOI: 10.1117/12.2185509, San Diego, California, USA, 2015

Boris Khots, Dmitriy Khots, Lagrangian in Classical Mechanics and in Special Relativity from Observer’s Mathematics Point of View,

DOI 10.1007/s10701-015-9895-4,

Foundations of Physics ISSN: 0015-9018 (Print) 1572-9516 (Online),

Springer, March 2015, Heidelberg, Germany; New York, USA

Boris Khots, Dmitriy Khots,
Small deviations between classical and observer's mathematics point of view on quantum mechanics, 16 pp,

Fields Institute, Toronto, Canada, March 2015

Boris Khots, Dmitriy Khots,
Classical and Quantum Mechanics aspects from Observer’s Mathematics point of view,
Talk at the International Congress of Mathematicians, Seoul 2014, Proceedings of ICM2014.

Dmitriy Khots, Boris Khots,
What is Behind Small Deviations of Quantum Mechanics Theory from Experiments?
Observer’s Mathematics Point of View, American Institute of Physics, Melville, New York volume 1637, 491 (2014)

Dmitriy Khots, Boris Khots,
Photoelectric Effect from Observer’s Mathematics Point of View,
American Institute of Physics, Melville, New York, volume 1637, 487 (2014)

Boris Khots, Dmitriy Khots,
Lagrangian in Classical Mechanics and in Special Relativity from Observer’s Mathematics Point of View,

Digital Object Identifier (DOI) 10.1007/s10701-015-9895-4,

Foundations of Physics ISSN: 0015-9018 (Print) 1572-9516 (Online),

Springer, March 2015, Heidelberg, Germany; New York, USA

Boris Khots, Dmitriy Khots, Quantum theory from Observer’s Mathematics point of view,
10th HSTAM 2013 Congress Proceedings, Dynamics,

#9, 6 pp, ISBN: 124-7485-789679-567-3,
Technical University of Crete, Greece, 2013

Dmitriy Khots, Boris Khots,
Observer’s Mathematics approach to the Quantum Mechanics,
Talk at the International Conference "Nonlinear Mathematical Physics: Twenty years of JNMP",
Nordfjordeid, Norway, 2013.

Boris Khots, Dmitriy Khots, Observer’s Mathematics applications to the Quantum Mechanics,
The Royal Swedish Academy of Sciences, Physica Scripta, volume 2014, #T163, December 2014.

Dmitriy Khots, Boris Khots, Geometrical and Quantum Mechanical aspects in Observer’s Mathematics, Proceedings of the SPIE Optics + Photonics 2013 Conference “8832 - The Nature of Light: What are Photons? V”,

vol. 8832, pp 8832 ID-1 – 8832 ID-6, San Diego, California, USA, 2013

Boris Khots, Dmitriy Khots, The foundations of Quantum Mechanics in Observer’s Mathematics, Quantum Theory:
Reconsideration of Foundations - 6, AIP Conference proceedings, Volume 1508,

pp 407-411, Melville, New York, 2012

Boris Khots, Dmitriy Khots, Quantum Mechanics problems in Observer’s Mathematics, ICNPAA 2012, AIP Conference proceedings, Volume 1493,

pp 518-522, Melville, New York, 2012

Boris Khots, Dmitriy Khots, The basis of Quantum Mechanics in Observer’s Mathematics,
Abstract of the talk at the 5th Chaotic Modeling and Simulation International Conference (CHAOS2012), Athens, Greece.

Boris Khots, Dmitriy Khots, Probability in Quantum Theory from Observer’s Mathematics point of view,
FOUNDATIONS OF PROBABILITY AND PHYSICS - 6. AIP Conference Proceedings, Volume 1424, pp. 154-159 (2012).

Boris Khots, Dmitriy Khots, Observer’s Mathematics applications to Number Theory, Geometry, Analysis,
Classical and Quantum Mechanics,
Ученые Записки Казанского Университета, Серия Физико-Математические науки, 2011, Т. 153, кн. 3, С. 196-203

Boris Khots, Dmitriy Khots, Hamilton equations of general relativity in Observer’s Mathematics, Proceedings of 4th Chaotic Modeling and Simulation (CHAOS 2011)
International Conference, Agios Nikolaos, Crete Greece, pp 203-208, 2011

Boris Khots, Dmitriy Khots, Observer’s Mathematics possible applications to Logistic-Transportation systems, Talk at “Contemporary information technologies in logistic-transportation control systems” International Conference, International Informatization Akademy, Kazan, Tatarstan, Russia, November 2011

Boris Khots, Dmitriy Khots, Two-slit interference and wave-particle duality for single photons from Observer’s Mathematics point of view, SPIE Proceedings, vol. 8121, H1-H6, 2011

Boris Khots, Dmitriy Khots, Analogy of Hilbert’s ten problem in Observer’s Mathematics, Talk at the International Congress of Mathematicians, Hyderabad 2010, Proceedings of ICM2010

Dmitriy Khots, Boris Khots, Lehmer’s number in Observer’s Mathematics, Talk at "17th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (17th ICFIDCAA)", Ho Chi Min City University of Pedagogy, Vietnam, August 2010

Dmitriy Khots, Boris Khots, Chaos problems in Observer’s Mathematics, Chaos theory: Modeling, Simulations and Applications, pp 215 – 222, World Scientific, 2011

Dmitriy Khots, Boris Khots, Chaos from Observer’s Mathematics point of view, “Chaotic systems. Theory and applications”, pp 111-118, World Scientific, 2010

Dmitriy Khots, Boris Khots, Quantum Theory from Observer’s Mathematics point of view, American Institute of Physics, volume 1232, pp 294-298, Melville, New York, 2010

Dmitriy Khots, Boris Khots, Solitary Waves and Dispersive Equations from Observer’s Mathematics point of view, “Geometry “in large”, topology and applications”, pp 86-95, Kharkov, Ukraine,2010.

Dmitriy Khots, Boris Khots,
Physical Aspects of Observer’s Mathematics, American Institute of Physics, volume 1101, pp 311-313, Melville, New York, 2009

Dmitriy Khots, Boris Khots,
Non-Euclidean Geometry in Observer’s Mathematics, Acta Physica Debrecina, tomus XLII, pp 112-119, Debrecen, Hungary, August 2008

Dmitriy Khots, Boris Khots,
Data Mining in Observer’s Mathematics, International Journal of Pure and Applied Mathematics, volume 51, #2, pp 195-201, Bulgaria, 2009

Dmitriy Khots, Boris Khots,
Tenth Hilbert Problem in Observer’s Mathematics, Proceedings of the 16th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications (16th ICFIDCAA), pp 81-85, Dongguk University, Gyeongju, Korea, 2008

Dmitriy Khots, Boris Khots, Fermat’s, Mersenne’s and Waring’s
problems in Observer’s Mathematics, International Journal of Pure and Applied
Mathematics, Bulgaria, v. 43, #3, pp 403-408 (2008)

Dmitriy Khots, Boris Khots, Quantum Theory and Observer’s Mathematics, American
Institute of Physics (AIP), volume 962, pp 261-264, 2007.

Boris Khots, Dmitriy Khots, Observer’s Mathematics – Mathematics of Relativity,
Applied Mathematics and Computations, volume 187, issue 1, April 2007, pp
228-238, New York.

Boris Khots, Dmitriy Khots, An Introduction to the
Mathematics of Relativity, Lecture Notes in Theoretical and Mathematical
Physics, Ed. A.V. Aminova, Kazan State University, v. 7, pp 269-306, 2006.

Boris Khots, Dmitriy Khots, Physical Theory of Relativity and Mathematics of Relativity, Recent Problems in Field Theory, Ed. A.V. Aminova, Kazan State University, v. 5, pp 239-242, 2006.

Dmitriy Khots, Euclidean and Lobachevsky Geometries in Mathematics of Relativity, Recent Problems in Field Theory, Ed. A.V. Aminova, Kazan State University, v. 5, pp 243-246, 2006.

Boris Khots, Dmitriy Khots, Analogy of Fermat’s last problem in Observer’s Mathematics - Mathematics of Relativity, Talk at the International Congress of Mathematicians, Madrid 2006, Proceedings of ICM2006

Full text in .pdf you can see below.

An Introduction to the Mathematics of Relativity (first edition - 2004)

Title

Dedication and Thanks

AMS Classification

Chapter 0. Introduction

Chapter 1. Arithmetic

Chapter 2. Algebra

Chapter 3. Geometry

Chapter 4. Analysis & Topology

Chapter 5. Logic

Chapter 6. Einstein

Appendix 1 TABLE: V2 X_{2}
99.99 (nonnegative elements)

Appendix 2 TABLE: V2 X_{2} 99.98
(nonnegative elements)

Appendix 3 TABLE: V2 X_{2} 99.97
(nonnegative elements)

Appendix 4 TABLE: V2 X_{2} 99.95
(nonnegative elements)

Appendix 5 TABLE: V2 X_{2} 99.92
(nonnegative elements)

Appendix 6 TABLE: V2 X_{2} 99.90
(nonnegative elements)

Appendix 7 TABLE: V2 X_{2} 99.53
(nonnegative elements)

Appendix 8 TABLE 1: V2 X_{2} 99.99 +_{2}
V2 X_{2} 99.98

Appendix 9 TABLE 2: ((V2 X_{2} 99.99 +_{2}
V2 X_{2} 99.98) +_{2} V2 X_{2} 99.97) (except data from
Table 1)

Appendix 10 TABLE 3: (((V2 X_{2}
99.99 +_{2} V2 X_{2} 99.98) +_{2} V2 X_{2}
99.97) +_{2} V2 X_{2} 99.95) (except data from Tables 1, 2)

Appendix 11 TABLE 4: ((((V2 +_{2}
99.99 +_{2} V2 X_{2} 99.98) +_{2} V2 X_{2}
99.97) +_{2} V2 X_{2} 99.95) +_{2} V2 X_{2}
99.92) (except data from Tables 1,2,3)

Appendix 12 TABLE 5: (((((V2 X_{2}
99.99 +_{2} V2 X_{2} 99.98) +_{2} V2 X_{2}
99.97) +_{2} V2 X_{2} 99.95) +_{2} V2 X_{2}
99.92) +2 V2 X_{2} 99.90) (except data from Tables 1,2,3,4)

Appendix 13 TABLE 6: ((((((V2 X_{2}
99.99 +_{2} V2 X_{2} 99.98) +_{2} V2 X_{2}
99.97) +_{2} V2 X_{2} 99.95) +_{2} V2 X_{2}
99.92) +_{2} V2 X_{2} 99.90) +_{2 }V2 X_{2}
99.53 (except data from Tables 1,2,3,4,5,6)

Appendix 14 Intersecting Lines on Plane –
Empty Intersection

Appendix 15 Intersecting Lines on Plane –
Intersection number = 1

Appendix 16 Intersecting Lines on Plane –
Intersection number = 2

Appendix 17 Intersecting Lines on Plane –
Intersection number = 10

Appendix 18 Intersecting Lines on Plane –
Intersection number = 100

Appendix 19 Riemannian Geometry: Intersecting
Planes on Cube – Empty Intersection

Appendix 20 Riemannian Geometry: Intersecting
Planes on Cube – Intersection number = 2

Appendix 21 Riemannian Geometry: Intersecting
Planes on Cube – Intersection number = 4

Appendix 22 Riemannian Geometry: Intersecting
Planes on Cube – Intersection number = 20

Appendix 23 Appendixes 14-22, Summary

© 2004 Boris Khots, Dmitriy Khots